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Vibrations and Waves

Section 3

What do you think?

• Consider different types of waves, such as water

waves, sound waves, and light waves. What

could be done to increase the speed of any one

of these waves? Consider the choices below.

• Change the size of the wave? If so, in what way?

• Change the frequency of the waves? If so, in what

• Change the material through which the wave is

traveling? If so, in what way?

Section 3

Wave Motion

• A wave is a disturbance that propagates through

a medium.

– What is the meaning of the three italicized terms?

– Apply each word to a wave created when a child

jumps into a swimming pool.

• Mechanical waves require a medium.

• Electromagnetic waves (light, X rays, etc.) can

travel through a vacuum.

Section 3

Wave Types

• The wave shown is a pulse wave.

– Starts with a single disturbance

• Repeated disturbances produce periodic waves.

Section 3

Wave Types

• If a wave begins with a disturbance that is SHM, the

wave will be a sine wave.

• If the wave in the diagram is moving to the right, in which

direction is the red dot moving in each case?

Section 3

Transverse Waves

• A wave in which the particles move perpendicular to the

direction the wave is traveling

• The displacement-position graph below shows the

wavelength () and amplitude (A).

Section 3

Click below to watch the Visual Concept.

Visual Concept

Transverse Wave

Section 3

Longitudinal Wave

• A wave in which the particles move parallel to the

direction the wave is traveling.

– Sometime called a pressure wave

• Try sketching a graph of density vs. position for the

spring shown below.

Section 3

Visual Concept

Longitudinal Wave

Section 3

Wave Speed

• Use the definition of speed to determine the speed of a

wave in terms of frequency and wavelength.

– A wave travels a distance of one wavelength () in

the time of one period (T), so

– Because frequency is inversely related to period:

Section 3

Wave Speed

• SI unit: s-1 m = m/s

• The speed is constant for any given medium.

– If f increases, decreases proportionally.

– Wavelength () is determined by frequency and speed.

• Speed only changes if the medium changes.

– Hot air compared to cold air

– Deep water compared to shallow water

Section 3

Visual Concept

Characteristics of a Wave

Section 3

Waves Transfer Energy

• Waves transfer energy from one point to another

while the medium remains in place.

– A diver loses his KE when striking the water but the

wave carries the energy to the sides of the pool.

• Wave energy depends on the amplitude of the

– Energy is proportional to the square of the amplitude.

• If the amplitude is doubled, by what factor does the energy

increase?

• Answer: by a factor of four

Section 3

Now what do you think?

• Consider different types of waves, such as water

waves, sound waves, and light waves. What

could be done to increase the speed of any one

of these waves? Consider the choices below.

– Change the size of the wave? If so, in what way?

– Change the frequency of the waves? If so, in what

– Change the material through which the wave is

traveling? If so, in what way?

Section 4

What do you think?

• Imagine two water waves traveling toward each other in a swimming pool. Describe the behavior of the two waves when they meet and afterward by considering the following questions. • Do they reflect off each other and reverse direction?

• Do they travel through each other and continue?

• At the point where they meet, does it appear that only one wave is present, or can both waves be seen?

• How would your answers change for a crest meeting a trough?

Section 4

Wave Interference

• Superposition is the combination of two

overlapping waves.

– Waves can occupy the same space at the same time.

– The observed wave is the combination of the two

waves.

– Waves pass through each other after forming the

composite wave.

Section 4

Constructive Interference

• Superposition of waves that produces a resultant wave

greater than the components

– Both waves have displacements in the same direction.

Section 4

Destructive Interference

• Superposition of waves that produces a resultant wave

smaller than the components

– The component waves have displacements in opposite

directions.

Section 4

Visual Concept

Comparing Constructive and Destructive

Interference

Section 4

Reflection: Free End

• The diagram shows a

wave reflecting from an

end that is free to move

up and down.

• The reflected pulse is

upright.

– It is produced in the same

way as the original pulse.

Section 4

Reflection: Fixed End

• This pulse is reflected

from a fixed boundary.

• The pulse is inverted

upon reflection.

– The fixed end pulls

downward on the rope.

Section 4

Standing Waves

• Standing waves are produced when two

identical waves travel in opposite directions and

interfere.

– Interference alternates between constructive and

destructive.

• Nodes are points where interference is always

destructive.

• Antinodes are points between the nodes with

maximum displacement.

Section 4

Standing Waves

• A string with both ends fixed

produces standing waves.

– Only certain frequencies are

possible.

• The one-loop wave (b) has

a wavelength of 2L.

• The two-loop wave (c) has

a wavelength of L.

• What is the wavelength of

the three-loop wave (d)?

– 2/3L

Section 4

Visual Concept

Standing Wave

Section 4

What do you think?

• Imagine two water waves traveling toward each other in a swimming pool. Describe the behavior of the two waves when they meet and afterward by considering the following questions. • Do they reflect off each other and reverse direction?

• Do they travel through each other and continue?

• At the point where they meet, does it appear that only one wave is present or can both waves be seen?

• How would your answers change if it was a crest and a trough?

Section 1

What do you think?

• What is sound? • What do all of the sounds that you hear have in

common?

• How do they differ?

• Can sounds travel through solids? Liquids? Gases? • Is one type of material better for transmitting sound

waves?

• When race cars or emergency vehicles pass you, the sound changes. In what way, and why?

Section 1

What is Sound?

• Sound is a longitudinal wave.

• All sound waves are produced

by vibrating objects.

– Tuning forks, guitar strings, vocal

cords, speakers

• The vibrating object pushes the

air molecules together, forming

a compression.

• It then spreads them apart,

forming a rarefaction.

Section 1

Graphing Sound Waves

• The diagram shows

compressions (dark) and

rarefactions(white). If you

measured the pressure or

density of the air and

plotted these against

position, how would the

graph appear?

Section 1

Visual Concept

Sound Waves

Section 1

Characteristics of Sound

• Frequency is the number of waves per second.

• You have heard of ultrasound. What is it?

• Frequencies audible to humans are between

20 Hz and 20 000 Hz.

– Middle C on a piano is 262 Hz.

– The emergency broadcast signal is 1 000 Hz.

• Infrasound frequencies are lower than 20 Hz.

• Ultrasound frequencies are greater than

20 000 Hz.

Section 1

Visual Concept

Comparing Infrasonic and Ultrasonic Sounds

Section 1

• Pitch is the human perception of how high or low

a sound appears to be.

– Pitch is primarily determined by frequency.

– Pitch also depends slightly on other factors.

• Higher frequencies appear to have a higher pitch when

played loudly, even though the frequency does not change.

Section 1

Speed of Sound

• Sound waves travel through solids, liquids and

gases.

– In which would the speed generally be greatest?

• Solids. Because the molecules are more closely packed, the

particles respond more rapidly to compressions.

– How might the temperature of air affect the speed of

sound waves? Why?

• Higher temperature increases the speed of the waves

because the particles are moving faster and colliding more

often.

Section 1

Speed of Sound

Section 1

Spherical Waves

• Sound propagates in three dimensions.

• The diagram shows: – Crests or wave fronts (blue

circles)

– Wavelength ()

– Rays (red arrows)

• Rays indicate the direction of propagation.

• How would these wave fronts appear different if they were much farther from the source?

Section 1

Spherical Waves

• Wave fronts and rays become

more nearly parallel at great

distances.

• Plane waves are simply very

small segments of a spherical

wave a long distance from the

source.

Section 1

Doppler Effect

• Why are the waves closer together on the left?

– Waves are closer because the vehicle moves to the left

along with the previous wave.

• How will the

sound be different

for observer A and

observer B?

– Higher frequency

(pitch) for

observer A

• Continued on the next

slide….

Section 1

Doppler Effect

• How would the wave

pattern change if the

vehicle moved at a faster

speed? How would it

sound different?

– At a higher speed, waves

would be even closer

together and the pitch

difference would be even

greater.

• The Doppler effect is the observed change in

frequency due to the motion of the source or observer.

Section 1

Visual Concept

Doppler Effect and Sound

Section 1

• What is sound? – What do all of the sounds that you hear have in

common?

– How do they differ?

• Can sounds travel through solids? Liquids? Gases? – Is one type of material better for transmitting sound

waves?

• When race cars or emergency vehicles pass you, the sound changes. In what way, and why?

Section 2

What do you think?

• Members of rock bands generally protect their

ears from the loud sounds to prevent damage to

their hearing.

• How do we determine the loudness of a sound?

• What quantity is loudness measuring?

• What units are used?

• Name three ways you can reduce the loudness of the

music heard by a person in the audience.

Section 2

Sound Intensity

• Vibrating objects do work on the air as they push

against the molecules.

• Intensity is the rate of energy flow through an

– What is “rate of energy flow” called?

• E/t is called power (P).

– Since the waves spread out spherically, you must

calculate the area of a sphere. How?

• A = 4r2

– So, what is the equation for intensity?

Section 2

Sound Intensity

• SI unit: W/m2

• This is an inverse square relationship.

– Doubling r reduces intensity by ¼.

– What happens if r is halved?

• Intensity increases by a factor of 4.

Section 2

Intensity and Decibels

• An intensity scale based on human perception of

loudness is often used.

• The base unit of this scale is the bel. More

commonly, the decibel (dB) is used.

– 0.1 bel = 1 dB,1 bel = 10 dB, 5 bels = 50 dB, etc.

– The lowest intensity humans hear is assigned a value

of zero.

• The scale is logarithmic, so each increase of 1

bel is 10 times louder.

– An increase in intensity of 3 bels is 1 000 times louder.

Section 2

Classroom Practice Problems

• The intensity of the sound from an explosion is

0.10 W/m2 at a distance of 1.0 × 103 m. Find the

intensity of the sound at a distance of 5.0 × 102

m, 1.0 × 102 m and 10.0 m.

– Answers: 0.41 W/m2, 1.0 × 101 W/m2, 1.0 × 103 W/m2

• Find the approximate decibel equivalents of

these sound intensities using Table 2.

– Answers: 110 dB, 130 dB, 150 dB

Section 2

Audible Sounds

• The softest sound humans can hear is called the

threshold of hearing.

– Intensity = 1 10-12 W/m2 or zero dB

• The loudest sound humans can tolerate is called

the threshold of pain.

– Intensity = 1.0 W/m2 or 120 dB

• Human hearing depends on both the frequency

and the intensity.

Section 2

Visual Concept

Human Hearing

Section 2

Forced Vibrations

• Sympathetic vibrations occur when a vibrating

object forces another to vibrate as well.

– A piano string vibrates the sound board.

– A guitar string vibrates the bridge.

• This makes the sound louder and the vibrations

die out faster.

– Energy is transferred from the string to the sound

board or bridge.

Section 2

Resonance

• The red rubber band links the 4

pendulums.

• If a blue pendulum is set in

motion, only the other blue

pendulum will have large-

amplitude vibrations.

– The others will just move a small

amount.

• Since the vibrating frequencies

of the blue pendulums match,

they are resonant.

Section 2

Resonance

• Large amplitude vibrations produced when the

frequency of the applied force matches the

natural frequency of receiver

– One blue pendulum was the driving force and the

other was the receiver.

• Bridges have collapsed as a result of structural

resonance.

– Tacoma Narrows in the wind

– A freeway overpass during an earthquake

Section 2

Visual Concept

Resonance (Frequency)

Section 2

• Members of rock bands generally protect their

ears from the loud sounds to prevent damage to

their hearing.

– How do we determine the loudness of a sound?

• What quantity is loudness measuring?

• What units are used?

– Name three ways you can reduce the loudness of the

music heard by a person in the audience.

Section 3

What do you think?

• A violin, a trumpet, and a clarinet all play the

same note, a concert A. However, they all sound

different.

• What is the same about the sound?

• Are the frequencies produced the same?

• Are the wave patterns the same?

• Why do the instruments sound different?

Section 3

Standing Waves on a String

• There is a node at each end because

the string is fixed at the ends.

• The diagram shows three possible

standing wave patterns.

• Standing waves are produced by

interference as waves travel in

opposite directions after plucking or

bowing the string.

• The lowest frequency (one loop) is

called the fundamental frequency (f1).

Section 3

Standing Waves on a String

• To the left is a snapshot of a single loop standing wave on a string of length, L.

• What is the wavelength for this wave? – Answer: = 2L

• What is the frequency? – Answer:

Section 3

Harmonics

• n is the number of loops or harmonic number.

• v is the speed of the wave on the string. – Depends on tension and density of the string

• L is the length of the vibrating portion of the string.

• How could you change the frequency (pitch) of a string?

Section 3

Visual Concept

Fundamental Frequency

Section 3

Standing Waves in an Air Column

• Wind instruments also use standing waves.

– Flutes, trumpets, pipe organs, trombones, etc.

• Some instruments have pipes open at both ends

while others have one end closed.

– Air is free to move at open ends so antinodes occur.

– Closed ends are nodes.

• The velocity of the wave is now the velocity of

sound in air (346 m/s at 25°C).

Section 3

Both Ends Open

Section 3

Closed at One End

Section 3

Wind Instruments

• Wind instruments are not as simple as organ

pipes.

– The shape is not always cylindrical.

– The holes change the wave patterns as well.

– The size of the “pipe” varies along the length.

Section 3

• One string on a toy guitar is 34.5 cm long.

– What is the wavelength of the first harmonic or the

fundamental wavelength?

• Answer: 69.0 cm or 0.690 m

– The string is plucked and the speed of the waves on

the string is 410 m/s. What are the frequencies of the

first three harmonics?

• 590 Hz, 1200 Hz, 1800 Hz

• Note: The use of significant figures causes the multiples of

590 to be 1200 and 1800 because only two significant figures

are present in the answer.

Section 3

• An organ pipe open at both ends is 34.5 cm long. – What is the wavelength of the first harmonic or the

fundamental wavelength? • Answer: 69.0 cm or 0.690 m

– What are the frequencies of the first three harmonics if the air temperature is 25.0°C?

• Answers: 501 Hz, 1000 Hz, 1500 Hz

– Answer the same questions if the pipe is closed at one end.

• Answers: 251 Hz, 753 Hz, 1250 Hz

Section 3

Timbre or Quality of Sound

• Instruments do not vibrate in a single mode.

– Several harmonics are produced at the same time.

– The particular harmonics and intensity of each vary

with different instruments.

• Timbre is the quality of the tone resulting from

the combination of harmonics.

• The fundamental frequency (1st harmonic)

determines the pitch.

– Adding other harmonics changes the timbre.

Section 3

Timbre

Section 3

Visual Concept

Timbre

Section 3

• The diagram

shows two waves

of different

frequencies.

Sketch the

superposition or

sum of these

waves.

• How would the

combined wave

sound?

Section 3

• Produced by two waves with the same intensity and different frequencies – Generally the frequencies are nearly the same.

• The sound pulses or changes from loud to soft and back.

• Beats are used to tune instruments. – If a tuner hears beats, the instruments are slightly out

of tune.

• The number of beats heard per second is the difference in the two frequencies.

Section 3

Visual Concept

Section 3

• A violin, a trumpet, and a clarinet all play the

same note, a concert A. However, they all sound

different.

– What is the same about the sound?

• Are the frequencies produced the same?

• Are the wave patterns the same?

– Why do the instruments sound different?

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